Theory of thermionic emission

469

Surface Science I I5 (I 982) 469-500 North-Holland Publishing Company

THEORY OF THERMIONIC

EMISSION

A. MODINOS Depurtmenrof Electricul Received

Engineering,

16 July 1981; accepted

lJnioersi[v

for publication

of Sal&-d,

Salford

8 December

MS

4 WT,

L/K

1981

A comprehensive theory of thermionic emission from clean metal surfaces is presented. The theory takes into account the energy band structure of the metal, inelastic scattering due to electron-electron collisions and the thermal vibration of the atoms. We applied the theory to thermionic emission from Cu( 100). We calculated the thermally emitted current from this plane as a function of applied field. We find an almost periodic deviation from the Schottky line, similar in nature with that which is observed in emission from polycrystalline emitters [ 11. We believe that accurate measurements of the amplitude and phase of these deviations from the Schottky line can, when analysed in the manner described here, provide valuable information on the surface optical potential. We have also calculated the total energy distribution of the emitted electrons for a typical value of the applied field. The dependence of the above measurable quantities on the parameters which enter the theory is analysed and demonstrated by explicit numerical calculations.

1. Introduction Thermionic emission has a long history starting with the work of Richardson, Schottky and Von Laue [2] prior to the invention of quantum mechanics. Dushman [3] gave the first quantum mechanical theory of thermionic emission, but it is with the work of Fowler and Nordheim [4] that the theory was put on a firm foundation. Their theory was based on the proposition (Sommerfeld [5]) that the electrons inside a metal behave as if they were free particles. Following the discovery of small periodic deviations from the Schottky line (see section 5) by Seifert and Phipps [6], the theory of thermionic emission was reformulated in order to account for this phenomenon, by Guth and Mullin [7]. Further experimental work (always on polycrystalline emitters) on the periodic deviation effect [ 1,8] was followed by further theoretical work [9- 131 but always within the framework of the free-electron theory of Sommerfeld. Following the pioneering work of Nichols [ 141, a number of workers measured the thermionic current from a variety of single crystal planes. Most of these measurements have been concerned with the experimental determination of the work function on the basis of the Richardson-Schottky formulae (eqs. (12) and (76) and to a lesser degree with the determination of the pre-exponential term in Richardson’s formula (eq. (76)). A critical review of such measurements prior to 1955 has been given by Herring and Nichols [lo] and by Nottingham [8]. Work 0039-6028/82/0000-0000/$02.75

0 1982 North-Holland

470

A. Modinos / Theory

of thermionic emission

function measurements by thermionic and other methods have been reviewed by Riviere [ 15). A more general review of thermionic emission studies covering the years 1955 to 1970 has been given by Haas and Thomas [ 11. The above authors emphasised the utility of thermionic emission as a tool to investigate surfaces. However, the situation at present remains as follows. Except for the work of Stafford and Weber [16] on tungsten (111),, no reliable data on the “periodic deviations” from the Schottky line exist for single crystal planes. Reliable measurements of the pre-exponential term in Richardon’s equation (eq. (76)) do not exist either. Also, very few data exist on the energy distribution of thermally emitted electrons [ 171,in contrast to the considerable amount of data that now exists on the energy distribution of field emitted electrons [18]. On the theory side, although the significance of the energy band structure of the metal in thermionic emission has been discussed in a qualitative manner by Herring and Nichols [lo) and in a semi-quantitative way by Haas and Thomas [I]; a quantitative theory of thermionic emission which takes band structure effects properly into account has not been produced. It is the aim of the present paper to introduce such a theory, which is long overdue. The proposed theory takes into account, under certain assumptions, “inelastic” electron scattering due to many-body effects, and the thermal vibration of the atoms, as well. Sections 2 to 5 are devoted to a summary of the basic formulae, a description of the optical potential representing the forces acting on an electron in a real metal, and of the means of calculating the total energy distribution and the current density of the thermally emitted electrons. In section 6 the theory is applied to Cu( 100). The dependence of the measurable quantities (total energy dist~bution, deviations from the Schottky Ii& in the current versus field curve) is analysed and demonstrated by explicit numerical calculations. Admittedly, copper is not the ideal thermionic emitter because of its relatively low melting point (1356 K) which limits the working temperature region to 800- 1lOQK or so. (As far as we know thermionic emission data exist only for polycrystalline copper emitters [19].) We have chosen Cu( 100) to demonstrate the theory for two reasons. More is known from independent experiments about this surface (20,211 than any other. Secondly, it is a very interesting plane to study. This is so because of the energy gap which exists around the vacuum level along the TX symmetry direction (normal to this plane) which means that thermal electrons do not penetrate into the metal and for that reason the energy distribution and the current density which one measures in a thermionic emission experiment reflect strongly the properties of the optical potential in the surface region. The same may be true of other surfaces and it is, of course, true to some degree for any emitting surface. Naturally, this, or any other, method for the analysis of thermionic measurements can be useful in practice, only when reliable data are there to analyse. IIopefully the knowledge that such measurements can be analysed in a systematic way will provide additional stimulus for perfor~ng these experiments.

A. Modinos / Theory of thermionic emission

471

2. Thermionic emission: basic formulae Consider a semi-infinite metal extending from z = - cc to z = 0 and let the metal-vacuum interface (the plane z = 0) be parallel to a crystallographic plane. At sufficiently high temperatures, a small but significant number of electrons have energies higher than the maximum of the potential barrier at the surface and some of them escape from the metal giving rise to a thermionic current. The relevant energies are shown in fig. 1. E, denotes the Fermi level, V,,,, the barrier maximum and $I the workfunction for the given surface. For the moment we need not specify the forces acting on an electron inside the metal or in the immediate vicinity of the metal-vacuum interface. On the vacuum side, at distances larger than 4 A or so from the interface, the electron moves in a potential field adequately described by classical electrostatics. We have V(z)=(-e2/4z)

-eFz+EF

+$.

(1)

The first term represents the well known image potential energy and the second term is due to an electric field F which is applied to the surface to carry away to an appropriate collector the thermally emitted electrons. This potential barrier is shown schematically by the solid line in fig. 1. The maximum of the barrier occurs at z, ==( e/4F)“2 = 1 .9/F”2

vi

@a) (A),

(2b)

”

r--+f+--_---v.., # -I / II ‘\. t I I

L

IT\ nai I

EF

-Z
1

\.

1

I

If )‘mII Zn

\.

= Z-

Fig. i. The surface potential barrier (schematic). The solid line represents the actual barrier. The broken curve represents the parabolic approximation to the potential in region III. The dash-dot curve represents an assumed potential (non-reflecting) in region IV ( z’zm).

412

A. Modinos / Theory of thermionic emission

and has the value Vmax =F,

++-A+,

(3)

A+=(e3F)“*

(4a)

= 3.79F’/*

(eV).

(4b)

In eqs. (2b) and (4b), F is expressed

in V/A.

In a typical experiment

6.5x10-5
(5)

therefore, 24O>z,

~25

A,

(6)

0.03 < A+ -=z0.3 eV.

(7)

We note that the potential barrier described by eq. (1) decreases smoothly to the right of the barrier maximum (region IV, fig. l), and therefore the probability that an escaping electron will be reflected backward towards the metal, after leaving region III, is negligible. This means that the detailed shape of the potential in region IV, as long as it is a slowly varying, not reflecting potential, does not affect in any way the emitted current. We assume, without loss of generality, that the potential barrier reduces slowly to a constant value (zero) at some distance from the barrier maximum, in the manner indicated schematically by the dash-dot curve in fig. 1. Herring and Nichols [lo] have shown that the number of electrons which flow out of the metal (are emitted) per unit area per unit time with (total) energy between E and E + d E is given by j(E) d E (we refer to j(E) as the energy distribution of the emitted electrons), where j(E)

=

-r(E,k,,)]

=I[1 (2?rJ3h

d’k,,,

O
<(2mE,‘fi*)“*.

Here, f(E) is the Fermi-Dirac distribution function (the factor of 2 takes into account spin degeneracy), and r( E, k ,,) is the probability that a (free) electron moving from z = + cc towards the metal, with energy E and wavevector (parallel to the surface) k ,,, will be reflected, either elastically or inelastically at the metal surface. Using the fact, obvious from fig. 1, that only electrons with energy near or above the barrier maximum contribute to the thermionic current, one can easily show that + - (e3F)“* Ad=&

ev

k T B

)

exp( -&)j/(l

-+k,,))

d%? (9)

where c is defined c=E-V

rnax

by (10)

A. Modinos / Theory of thermionic emission

The limit of the k,, integration integrand in eq. (10) vanishes

413

need not be specified accurately because the except within a very limited region around

k,, =O.

The thermionic

(charge)

current

density

is given by

(11) where e is the magnitude (1 l), we obtain

of the electronic

~#2 - (e3F)“2

J=AiT’exp

-

kaT

i

charge.

Substituting

eq. (9) into eq.

I

(12)

3

where A = emki/2a2A3 is a universal i(T,F)=&

= 120 A cmP2 K-*

constant

(13)

and i is an average transmission

(k,T)p2/t-[exp( -CC

coefficient

defined d’k,,]

-&)//[l-r(r,k,,)]

by

de. (14)

We note that the integrand in the above equation decreases exponentially in both energy directions, for E > 0 because of the exponential term, and for z < 0 because of the transmission coefficient (1 - r). The dependence of i on the applied field comes through r. In the original Richardson-Schottky theory of thermionic emission, it is assumed that the reflection coefficient r equals unity when the normal energy of the electron lies above the barrier maximum and that it is zero otherwise. This leads to the following formula for the energy distribution of the emitted electrons:

exp[-

I_ h(E)

=

“-~as~)“2)

rexp(

-&),

forr>O,

(15)

2&)j3

lo,

for .z< 0,

and to the well know Richardson-Schottky formula for the thermionic current density, given by eq. (12) with i= 1. In order to go beyond the Richardson-Schottky theory it is necessary to calculate in a systematic manner the reflection coefficient r(c, k,, ) for realistic models of the metal surface. 3. The one-electron potential In order to calculate r(e, k,,), we must first describe the interaction responsible for the reflection of the electron at the surface. Let us, to begin

474

A. Modims

/ Theoy

of rhermionic emission

with, assume that inside the metal (z < 0) the electron sees a potential of the muffin-tin type, and that this is identical with the corresponding potential in the infinite crystal right up to the metal-vacuum interface (at z = 0) where it is terminated abruptly. Although at the high temperatures used in thermionic emission experiments the lattice constant is larger (usually by 3% or so) than that at room temperature, we may assume that the radius of the muffin-tin spheres and the spherical potential within each sphere are practically the same with the corresponding quantities at zero temperature. We note that the latter are knwon, from self-consistent calculations of the energy band structure, for many metals [22]. At this stage, we disregard the thermal vibrations of the atoms which we shall take into account later. The muffin-tin atoms are grouped into layers with 2-dimensional (2D) periodicity parallel to the surface in the manner described by Pendry [23] and the metal-vacuum interface (the plane z = 0) is taken at a distance (id+ D,), where d is the interlayer distance and D, an adjustable parameter, above the centre of the top (surface) layer (see fig. 2.). We note that in the case of copper and other transition metals of interest to thermionic emission each layer consists of a single plane of atoms with one atom per unit cell. We further assume that the potential on the vacuum side of the interface (z > 0) depends only on z, and we approximate it by

V*(z) =E,

+I$-

e2

4(z+z,)

-eFz.

(16)

The Fermi level E, is measured from the constant potential between the muffin-tin spheres which is taken as the zero of the energy. The third term in

Fig. 2. A one-dimensional (schematic) representation interface. The wells represent atomic layers of muffin-tin imaginary part of the potential (different scale).

of the potential at the metal-vacuum atoms. The dash-dot curve represents the

A. Modinos / Theoty of thermionic emrssion

475

the above equation represents a modified image potential barrier. z,, is an adjustable parameter which determines the magnitude of the potential in the immediate vicinity of the interface. Usually z0 = 0.5 A or so. For z ~z,,, V,(z) is practically identical with the barrier described by eq. (1) and therefore eqs. (2)-(4) which give the position and magnitude of the barrier maximum remain valid to a very good degree of approximation. A real one-electron potential field of the type we have just described, gives, at best, the force experienced by an electron in the average field (electrostatic plus exchange) due to all other electrons and nuclei in the metal. The difference between the actual (many-body) electron-electron interaction and the part of it represented in the above one-electron potential leads to “electron collisions” as a result of which a (primary) electron in an “excited one-electron state” may scatter into another such state of lower energy releasing energy which is used to excite another electron (a secondary electron) from an occupied state into an initially empty state with higher energy. Let us now consider the effect of the above mentioned inelastic collisions on a thermal electron incident on the metal from the right (see fig. 1). The energy of such an electron lies within a narrow region, 1E - V,, 15 k,T, around the barrier maximum. Since V,, - E, = 4 eV or so in a typical emitter, it is reasonable to assume that the vast majority of inelastic collisions will lead to a loss of energy by the incident (primary) electron much larger than kBT, which means that an inelastically scattered electron cannot escape from the metal for its energy will lie below the barrier maximum at z,. It is also obvious that secondary electrons generated by the above collisions will not in general have enough energy to escape from the metal. We conclude that at low incident energies (E = 0), the total reflection coefficient is identical with the elastic reflection coefficient. We note that measurements [21] of very low energy electron reflection at Cu(100) surfaces, confirm the truth of the above statement; Therefore, from now on r(tk,,), in eqs. (8)-( 14) and in what follows, will be identified with the elastic reflection coefficient. We emphasize, however, that inelastic collisions play a critical role in the determination of the elastic reflection coefficient. They lead to a reduction in this coefficient by removing electrons from the elastic channel. Fortunately, this can be taken into account within a one-electron potential, at least on a semi-empirical basis, by adding to the real potential, an imaginary component which acts as an absorber (a sink of electrons). One can easily show that when an electron with energy E and wavevector k is incident upon a scatterer represented by a finite region (e.g. a sphere) within which the potential is complex, the outgoing flux, calculated in the usual manner, is less than the incoming flux, and that the difference (the absorbed flux) is proportional to the imaginary component of the potential. The complex potential, sometimes referred to as the optical potential, will be denoted by

V(r) = V,(r) + i y,( E, r),

(17)

416

A. Modinos / Theory

of thermionic emission

where i = m. For a general discussion of the optical potential the reader is referred to Pendry’s book [23]. It is argued there that, in the bulk of the metal vi, depends on the energy of the electron but not (to a good approximation} on its position. It is usually further assumed that the imaginary part of the potential retains its bulk value right up to the metal-vacuum interface, diminishing rapidly to zero on the vacuum side of it. Following McRae and Caldwell [21], we parametrise Vi,( E, r) as follows

v,m(E’r)= y,(E)=

i

v,,(E),

forz
vi,(E)exp(-(z/p)*),

forz>O,

-a(/E-EF])7,

a>O.

(18) (19)

We emphasize that a and y are bulk parameters and thus independent of the surface under consideration. They can be determined from optical data for the given metal as shown by McRae [24]. We note, however, that in practice extracting the values of a and y from such data is not free of ambiguities. The constant p in eq. (18) determines the effective range of the imaginary component of the potential on the vacuum side of the interface and is, for practical purposes, an adjustable parameter. z0 in eq. (16) and the distance (id + DS) of the metal-vacuum interface (the plane z = 0) above the centre of the top layer are additional adjustable parameters. We assume that the interlayer distance d does not vary near the surface and that the potential between the muffin-tin spheres of the top layer and the interface plane at z = 0 is constant and equal to the interstitial potential in the bulk of the metal (the real part of this potential is by definition equal to zero). It is also assumed that the workfunction + is known.

4. Theory

4.1. The Herring-Nichols formuia for r(E, k ,,) We separate space into the five different regions marked I, II”, II@, III and IV in fig. 1. Regions II@ and IV are non-reflecting regions. In these regions the wavefunction of an electron with normal energy in the region of the barrier maximum can be adequately described by the WKB formulae (see, e.g., ref. [25]). In region II@, such an electron with energy E and wavevector k,, , propagating in the negative z direction, is described by *E.k,,(r) = exp(ik,,

lr,,> &(z),

(20)

(21)

A. Modinos /

Theory of thermionic emission

A2k2

E-$-

477

(22)

In region II@ the imaginary part of the potential is zero and E - Azki/2m V(z) > 0, so that q(z) in eq. (21) is real and positive. We have exp(ik,,

l

r,, ) $+,,,( 2)

aexp(ik,,

(in region IIa)

•r~i)[+iJ~>

+&$&(r)](inregion

IV),

(23)

where & represents a wave propagating in the negative z direction (incident on the barrier), and its complex conjugate +i*, a wave propagating in the positive z direction (reflected wave). Obviously, /X0 I2 is the coefficient of reflection of an electron by the potential barrier in region III acting in isolation. An electronic wave, in region II@, propagating in the negative z direction can, equally well, be described as follows XL>),,=exp(ik,,

J&*‘(Z)= hexp(

or,,) XL-)(Z), ficq(z)

(24) dz).

(25)

Incident upon the potential field to the left of region IIs, the wave described by eq. (24) will be partly reflected by it and partly transmitted into it. The “‘reflected wave”, denoted by XL;),,, consists of a specularly reflected beam and a number of “diffracted beams”. (At this stage we disregard the thermal vibrations of the atom.) We have

(26) where (g} are the 2D reciprocal vectors.corresponding to the crystallographic plane (surface) under consideration. For thermal electrons (E = V,,), the normal energy of the electron, E- (ti2,/2m)(k,, -tg)‘, in any of the “diffracted” beams (g # 0) lies well below the barrier maximums, so x6;‘) = 0 in the outer part (large z) of region 118. It follows that (transmitted wave in I) =.

exp(ik

lq)[xb-‘(2)

+ hxb+Y~)f (27)

(in outer IIa), which can be rewritten as follows (transmitted wave in I) =

exp&

l~,,)[lc;,(z)

+cL~(J%&,) rexI@)

G(r)]

(in outer IIs), (28)

478

A. Modinos / Theory of thermionic emission

where n and 6 are, by definition, real numbers such that q exp(i8) = exp( 2iizmg( z) d.z

).

(29)

We note that q(z) is complex in region IIa because the potential is complex there. Combining eq. (28) with eq. (23), we finally obtain (transmitted wave in region I) *([l

+~~~exp(i~)

lr,,)

Xexp(ik,,

G]

+in(‘)

+[b

+~~~exp(i@~

~~(~>]f

(in region IV).

(30)

The first term on the right of the above equation represents a wave with its source at z = + co propagating towards the metal (incident wave) and the second term represents the reflected wave propagating towards z = + co. It follows from eq. (30) that the elastic reflection coefficient r( E, k ,,) is given by ‘(E,~,,)=I(X~-EL)/(~+~FLX*)I~~

(31)

A=&exp(-id),

(32a)

c1=?Il”o.

(32b)

A formula of this type was first derived by Herring and Nichols (lo]. 4.2. Calculation of p. The potential at the interface is shown schematically in fig. 2. Let an electron with energy E and wavevector k,, in region IIB be incident on the metal. This electron will be partly reflected by the potential field in regions I and II” and partly transmitted into the metal. The complete wavefunction in region II (IIa i- IIB) is given by 9 = B< x$,-‘(z) exp(ik,, -r,,) -t-II: x$,+‘(z) exp(ik,\ St-,,) -I- x

&+xf;e)(;)

exp[i(k,,

+s>

-rii]-

(331

g+o The wavefunctions xL*‘( z) are defined as follows. They are the solutions of the Schrodinger equation (34) in region II, which satisfy the following boundary conditions. XL”‘{z) are given by eq. (25) in region 11s. We note that the same expression will describe x$,” in region II” as well, if the WKB approximation is valid in that region, otherwise x(‘“(z) in region II* must be obtained by numerical integration of eq. (34). For thermal electrons (total energy around the barrier maximum) x(+‘(z) (g+O) decrease and X’-)(Z) (g#O) increase exponentially in region II b as 2 R

A. Modinos /

Theory of thermionic emission

419

increases. The physical meaning of the various terms in eq. (33) has already been explained in the previous section. In the region of constant potential between the muffin-tin atoms of the surface layer and the interface at z = 0 the wavefunction has the following form \k = EAl R

K&F=

exp(iK,+

lr)

+A;

exp(iK,-

or),

(35)

(k,, +g,-[$[E-ivim(E)] -(k,, +g)2]“2))

where V,,(E) is the imaginary part of the potential inside the metal (z < 0). (The real part of the interstitial potential equals zero by definition.) We note that the same g vectors (n in number including g = 0) appear in eqs. (33) and (35). The coefficients At in eq. (35) are obviously related to the B,’ coefficients in eq. (33) since the wavefunction \k and its derivative are continuous functions. These relations are conveniently expressed in terms of the reflection and transmission matrix elements of region II” (this includes the potential step at z = 0 if such exists) as follows Bg” = T-+(k,, A;

=R--

+g)

lA;

( k,, +g)*A,t

lB,-

+R++(k,,

+g)

+T+-(k,,

+g)*B,-.

,

(37) (38)

Eq. (37) tells us that a wave in region IIB (B,” x,’ (z) exp(i[k ,, + g] rll )) which propagates or decays in the positive z direction derives from a wave in region I through region II*, and from a wave in (Ai exp(i Kg+ r)) partly transmitted region IIs ( BgP xi (z) exp(i[k,, + g] r,, )) partly reflected by the potential in region II”. A similar interpretation applies to eq. (38). T -+(k,, + g) and the other matrix elements, which depend also on the energy although this has not been explicitly denoted in eqs. (37) and (38), can be obtained easily, after x:(z) have been evaluated in region II”, by matching an appropriate wavefunction on the left (z = 0 - ) or right (z = 0 + ) of the interface to its corresponding expression on the other side of the- interface. In what follows it is convenient to use a matrix notation whereby A * and B k denote the column vectors of n elements, {AZ } and {B,’ } respectively, and T -+, T +-, etc. are n X n diagonal matrices with elements {T -+(k,, + g)}. {T + -(k,, + g}, etc., respectively. In addition to the above matrices we introduce the reflection matrix RC of the crystal, defined by l

l

l

A+=R’A-.

(39)

The matrix element Rz*,( E, k,, ) is the amplitude of the diffracted beam described by exp(i Kg+ r) due to an incident beam described by exp(iK,S r). We note that only the potential field in region I enters the calculation of the matrix RC. The combination of eqs. (37)-(39) leads after some algebra, to the following l

l

A. Modinos / Theo?

480

matrix

of thermionic emissio~l

equation

B+=@BM=

(40)

T-+(Z--CR--)-‘RCT+-+R++,

(41)

where Z is the unit n X n matrix. The reflection amplitude pO( E, k,,) is by definition

(eqs. (27) and (33)) given

by EL&~,,)

(42a)

=&+/B,= Mp=Q’=O (EJ,,)

-M,.

(42b)

The calculation of the diagonal matrices describing reflection by, and transmission through, region II” is straightforward (see section 4.4), therefore the problem of calculating CL,,reduces to that of calculating RC. This is a much more difficult but a well known problem, because R’ is exactly the same matrix which one needs to calculate in LEED theory, and there are by now well established procedures for calculating this matrix. In our calculation of the thermionic current from Cu(100) (section 6) we used the doubling layer method, described by Pendry [23] for calculating RC. For the benefit of the reader who is not familiar with the theory of LEED and in order to explain how the effect of the thermal vibrations can be taken into account (under certain assumptions), we describe below the main steps in such a calculation. 4.3. Calculation of R’ We view the semi-infinite crystal as a stack of identical layers of muffin-tin atoms with two-dimensional periodicity parallel to the given surface. We first evaluate the scattering matrix elements M8TB(E, k,,) of a single layer (a single plane of atoms in our case) which are defined as follows. Let a plane wave r) be incident on the layer from the left (Here we chose the origin of exp(iK,+ coordinates at the centre of the layer, i.e. the centre of one atom in this layer.) solution of the This will be scattered by the layer; so the corresponding Schriidinger equation outside the muffin-tin spheres will consist of the incident wave and a scattered wave given by l

kc =

zM&exp(iK,f

or),

forz)O,

R’ xM,,exp(iK,T

lr),

forz
I

(43)

The sum over g’ in the above equation includes (in practice) a finite (n) number of 2D reciprocal vectors, so A4 * are n X n matrices. The matrix elements can be expressed in terms of the phase shifts, denoted as usual by S,, which describe the scattering of a spherical wave of angular momentum

A. Modinos /

ti,/m by the spherically of the layer. We have [26,27,23]

X (I-X),;,f,;,,(i-I

Theory of thermionic emission

symmetric

Y,,(QK;))

muffin-tin

exp(i8,)

481

atom, and by the geometry

sins,.

(44)

Here D is the area of the unit cell of the layer. q,,, is a spherical harmonic and QK; denotes the angular direction of K,’ (for a summary of the properties of spherical harmonics for complex K,. see, e.g. refs. [23,29]). Kg: denotes the z component of Kg, and K,=$ I denotes

($[E

-iV,,(E)])

the unit matrix,

Xlm;I”m” = i exp(i8,)

I’*. and the X matrix

2 exp[ik,, rA+rr)

l

is given by

(rk - rs)] G,,;v,fp(r,

- rk),

(45)

where 5 denotes the position of the s th atom in the layer; ( rk - rs) is a vector of the two-dimensional lattice associated with the layer and as such lies in the plane of the layer; G,,;,.,,,,(R)

= 2 4a( - 1)(‘-‘-Y”2( I’m’ x I.&@,)

B,,( I’m’; l”m”) = /&,.(a)

B,,(I’m’; Y,,(O)

- l)“‘+““/$‘(

z&R)

/“M”), Y,,,_,,,,,(52) dS1,

(46) (47)

and hj’)(z) denotes the spherical Hankel function which for large z has the asymptotic form associated with a scattered (outgoing) wave. We need not describe here the way one goes about to evaluate the various quantities which enter eq. (44). We would like to mention, however, that for computational speed the summation over lattice points in eq. (45) is, in practice, transformed (in the manner of Ewald [29]) into two sums, one over the real lattice and the other over the reciprocal lattice. The required formulae have been derived by Kambe [30] and a program evaluating both the X matrix elements and the MgTg matrix elements has been written by Pendry [23]. It is obvious that the potential field of the layer is symmetric about the plane of the layer, so the transmission matrix M + and the reflection matrix M - for a plane wave incident on the layer from the right are exactly the same with those (eq. 44)) for an electron incident on the layer from the left. It is a matter of straightforward algebra to obtain the scattering matrix elements of a stack of identical layers (all parallel to the surface but displaced relative to each other by a

482

A. Modims

/ The-my of therntiontc mttsston

vector a) once the scattering matrix elements of a single layer have been evaluated. For that purpose the waves on the right of a layer are first expressed with respect to a “right” origin at a point fu from the centre of this layer and the waves on the left of the layer with respect to a “left” origin at a point - +u from the centre of the layer. The transmission matrix (left - right and right -+ left; the two are not the same any more) and the reflection matrix (left - left, right -+ right) with respect to the new “origins” can easily be obtained in terms of the M,‘, elements. (We note that for a pair of consecutive layers, the “right” origin of the left layer coincides with the left origin of the “right” layer.) It is no more than a simple exercise to obtain the scattering of the two layers (considered as one entity) with the waves on the right of the pair referred to the “right” origin of the right layer and those to the left of the pair referred to the “left” origin of the left layer. In exactly the same way we use the scattering matrix elements of a pair of layers to find those of a pair of pairs of layers. We repeat the process until the reflection matrix of the slab is practically identical with the reflection matrix R’ of the semi-infinite crystal. The incident and diffracted waves obtained in the above manner refer to the “right” origin of the top layer (half an interlayer distance above the centre of the top layer). They can be referred (by a simple transformation) to a new origin, a distance ($d + 0,) from the centre of the top layer, if that is where the metal-vacuum interface (the plane z = 0) has been placed. (We recall that the potential between the muffin-tin atoms and the metal-vacuum interface is constant and equal to the bulk interstitial potential.) At this stage we are ready to consider the thermal vibration of the atoms and its effect on the scattering of the electron by the metal. When the atoms vibrate the perfect 2D periodicity of the metal parallel to the surface is destroyed. Therefore, when a beam of electrons with energy E and wavevector k,, is incident on the metal surface, the backscattered electrons are not restricted to a discrete set of diffracted beams {E, k,, +g). We have seen that in thermionic emission only electrons with energies around the barrier maxithe mum and k,, -0, matter and that, in the absence of thermal vibrations, reflection coefficient r( E, k ,,) is entirely determined by the amplitude of the specularly reflected beam. Now, diffuse scattering, i.e. scattering into angles other than those of the diffracted beams, leads to a reduction in the flux of the specularly reflected beam (and of any other diffracted beam). If we assume that the electrons off the specular direction, or at least the vast majority of them, can not escape from the metal because their normal energy lies below the barrier maximum, we can incorporate the effect of thermal vibration into our theory simply be calculating the effect that diffuse scattering has on the amplitude of the specularly reflected beam. We note, however, that the reduction in the reflection coefficient r( E, k ,,) calculated in the above manner may be slightly exaggerated (some electrons off the specular beam may escape from the metal). Fortunately, for thermal electrons the whole effect is relatively small, even at very high temperatures, and for that reason a more sophisticated

A. Modinos / TheoN of thermionic emission

483

treatment is not really necessary. We note in this respect that LEED experiments [31] have indeed shown that there is little diffuse scattering at low incident energies even at very high temperatures. The problem of how to incorporate the thermal vibration of the atoms in the calculation of the amplitude of the (coherent) diffracted beams {k ,, + g} has been solved by LEED theorists [32,33,23] under certain assumptions. These and the essential results can be summarised as follows. Consider the scattering of an electron with energy E and wavevector k,, by a single muffin-tin atom. Scattering into an angular direction specified by k’ (1k’ I= 1k I) is proportional to I4k’, t(k’,

k) I27

k) = 8a2 x t(f ‘m’; IM)( - l>“‘r,._,@,,)

y,,( n/J,

(48)

Im I’m’

t(I’m’; Im) = -S,,S,,,,,(2mE/A2)-’

exp(i6,)

sin a,,

(49)

where a,, is the usual Kronecker delta. We note that the scattering amplitude given by eq. (48) refers to an origin at the centre of the atom. If we keep the origin fixed and displace the atom by AR, t(k’, k) changes by a phase factor as follows t(k’, k) -+ t(k’, k) exp[i(k-

k’) *AR].

(50)

At high temperatures the atom vibrates around a mean position. However, because the atom moves slowly compared to the electron, it may be assumed stationary during the scattering process. In that case scattering of the electron by the vibrating atom (at temperature T) is described by a scattering amplitude t(T; k’, k) which is the average over displacements at temperature T of the quantity on the right hand side of eq. (50). It can be shown that [34] (exp[i(k-k’)*AR]). where (. . . ) denotes brackets. Hence

=exp(

-f([(k-k’)*AR]2)T],

the average over displacements

(51) of the quantity

within

the

t( T; k’, k) = r( k’, k) exp( -M),

(52)

M=

(53)

f([(k-k’)*AR12),.

The factor exp(-2M) is known as the Debye-Waller factor. The above derivation ignores energy exchange (of the order of k,T) between the electron and the vibrating atom so that Ikl = 1k'l = (2mE/h2)‘/2 in the above formulae (for thermal electrons E - E, = + = 4 eV B k,T). For a simple crystal (one atom per unit cell) at high temperatures (Ts O), one obtains the following approximate formula for M (see, e.g. ref. [23]) M=(lk-k’12,

(54)

5 = 3A2T/2m,kB02,

(55)

A. Modinos / Theory of thermionic emission

484

where mA is the mass of the atom and 0 is the Debye temperature of the solid which constitutes a measure of the compressibility of the solid. Eqs. (54) and (55) describe, at best, the vibration of a bulk atom. We expect the vibration of a surface atom to have on the average a larger amplitude than that of the bulk atom. Since the mean amplitude of the vibration is inversely proportional to the Debye temperature, the above implies that the effective of value of 0 for a surface atom is in general smaller than the bulk value of 0. In the case of Cu( loo), we find that thermal electrons do not penetrate much into the crystal because of the existence of an energy gap around the vacuum level. For that reason in our numerical calculations (section 6) we have taken a value for 8 (in eq. (55)) equal approximately to 2/3 of its bulk value. It can be shown [33,23] that when eq. (54) is substituted in eq. (52) the resulting formula for t( T, k’, k) can be transformed into an expression which is formally identical with that given by eqs. (48) and (49) except that the phase shifts now depend on the temperature as follows exp(ia,(T)

sin S,(T))

= ,z,,i’

exp( -2[k2)

X exp(isY)

sin S,

j,,( -2

i.$k2)

4n(21’ + lf(21” 21+ 1

+ 1)

10), 1 BJ1’0; ‘/2

(56)

where k2 = (2mE)/A2, j,(z) is the spherical Bessel function, and B,,, is the integral of three spherical harmonics defined by eq. (47). A program which evaluates the above quantity, given the zero temperature phase shifts S,, can be found in ref. [23].. We note that S,(T) is a complex quantity with a positive imaginary part, which implies that the outgoing flux is less than the incident flux on the atoms. This is due to the fact that all flux scattered incoherently has been neglected in the above averaging procedure. It is obvious that,‘having an expression for the scattering amplitude by the individual (vibrating) atom in terms of temperature dependent phase shifts, which is formally identical with the corresponding expression at zero temperature (atom fixed at its mean position), allows us to use the same formula (eq. (44)) with 6, replaced by 6,(T), to obtain the scattering matrix for the entire layer at the given temperature. We must remember that the lattice constant (the lattice, at finite temperatures, gives the mean positions of the atoms) is different (larger) at higher temperatures. The reflection matrix I? for the semi-infinite crystal is obtained by the doubling layer method in the manner already described in section 4.2. 4.4. Calculation of the surface barrier scattering matrix elements Scattering by the surface barrier (z 2 0)is described by the scattering matrix elements associated with region II” (T _ + , T + -, R + +, I? --) and also by the

A. Modinorr /

Theory of thermionic emission

485

parameters X,, 9 and 8, which appear in eqs. (31) and (32). These quantities can always be calculated numerically for any surface barrier (which depends only on z). In fact, that may be necessary for an accurate quantitative analysis of a given set of data. At this stage, however, an approximate evaluation of the above parameters for the barrier described by eqs. (16)-( 19) based, at least partly on analytic formulae, should be sufficient. The evaluation of the transmission (T + f and T + - ) and reflection amplitudes (R - - and R + +) of region II* (defined by (37) and (38)) is greatly simplified if we assume that the WKB appro~mation remains valid in region II” (for z > 0) as well as in region 11s. This is not a bad approximation for the chosen potential barrier (eqs. (16)-(19)) and we have, in fact, used it in our numerical calculations. The corresponding explicit formulae for T - + , R --, etc. are straightforward and need not be given here. We should point out, however, that the detailed shape of the potential variation in region II” is not known, and that most of the previous work [9-13J on the periodic deviations from the Schottky line has primarily been directed at understanding the nature of this region by comparing theoretical models for this barrier with experiment. In these calculations, however, the energy-band structure, inelastic electron collisions and the thermal vibration of the atoms were neglected (see, also, section 6). For the purpose of evaluating X,, as defined by eq. (23) we assume, as in earlier theories [7,9,12], that the potential barrier in region III is adequately described by the first two non-zero terms of its Taylor expansion around z, (the position of the barrier maximum). We have (ZinregionIII), v(z)Kl, -2(eF3)“2(z-z,)2 (57) where Vi% denotes. the magnitude of the barrier ma~mum. The above, known as the parabolic approximation to the potential, is shown schematically by the broken curve in fig. 1. The corresponding Schrodinger equation can be put into a form known as Weber’s equation whose solutions, corresponding to different asymptotic behaviour as (z - z,,,) -+ c co, are readily available in the literature (see, e.g., ref. [35]). By constructing the solution which has the asymptotic behaviour dictated by eq. (23), we obtain h, = r*(+ - i&)(2n)-“2

exp( - &S - ia In 6 - fni),

6 = 8-‘/2(2m/ti2)“2(eF3)-“4~.

(58)

(59)

r(x) is the well known Gamma Function and the star denotes, as usual, complex conjugation; u is the normal energy of the electron relative to the barrier maximu, i.e. u=a-h2ki/2m.

(60)

One can easily show, starting from eq. (58) and using standard properties of the Gamma Function, that IX,]= (1 +e2ns)-“2.

(61)

A. Modinos / Theory of thermionic emission

486

We need, also, an expression for the argument of A,. We note that according to eq. (61) )A,, I= 1 for S 5 - 1 (normal energy of the electron a bit below the barrier maximum according to eq. (59)) in which case the reflection coefficient r( E, k ,,) is also equal to unity. Similarly 1A, ) = 0 for S Z. 1 (energy a bit above the barrier maximum). We therefore need an expression for arg{h,} only when 16I a 1. We have (see, e.g., ref. [36]) arg{r(f Cr

-is)}

for lf3jK 1,

y CS,

(62)

1.9635.

Substituting arg{h,}

(63) eq. (62) into eq. (58) we obtain

* - (S InISI +CS+

fir),

for 161K 1.

(64)

The coefficients n and 8, defined by eq. (29) can be calculated We note that V,,( E, z) is different from zero only in the immediate the interface (i.e. for z 5 /3), and that there ) Y,@,

~)/a

W-

as follows. vicinity of (65)

V,(z),

where W is the normal

energy of the electrons,

defined

by

W= E - h2ki/2m = v,,

(66)

+ u.

(67)

It follows from eq. (65) that q(z) is to very good approximation

given by

(68) Substituting

eq. (68) into eq. (29) we obtain ‘I2

V,,(E, [w-

EY2/Zm(

$[W-

z) dz v,(z)]“2

1’

V,(z)])“2dz.

(69)

(70)

0

The numerical evaluation of the integral computing time because of the very short reason no additional approximation need be In order to evaluate 8 we proceed as follows 0=2/=-(

$[W-

in eq. (69) requires very little range of V,,(E, z) and for that made in evaluating this quantity. [9]. We write

v,(z)])“‘dz

5 +2 L’( $[W-

I$(z)])“‘.dz,

(71)

A. Modinos / Theory of thermionic emission

487

where { lies at the border line between regions III and 11s. The first integral in eq.. (71) can be evaluated analytically by replacing V,(z) in the integrand by its parabolic approximation (eq. (57)). The second integral in eq. (71) we evaluate as follows. We note that an accurate evaluation of 6 is required only for small values of U, hence we replace W in the integrand by its equal according to eq. (67), we expand the integrand in Taylor series around u = 0 and keep only the first two terms of this series. The resulting integral can be performed analytically. After some calculation one obtains 8-

--6ln]S]--(g+C)S+y-(SmeZz,)“*/ft,

g=4-C-ln y =

(72)

12-lny,

(73)

+( me*/h*)“*( e/F)“*,

where S is given by eq. (59), C by eq. (63) and z0 is the adjustable eq. (16) for the potential barrier.

(74) parameter

in

5. Deviations from the Schottky line According to the original Schottky theory (eq. (12) with i= 1) a plot of ln(J/T*) versus fi must be a straight line (Schottky line) with slope m = e3/*/k,T.

(75)

In practice, small “periodic” deviations from the Schottky line (similar to those shown in fig. 8) have been observed in emission from polycrystalline emitters (see, e.g., ref. [l]) and from single crystal planes [16]. The physical origin of these deviations may be understood as follows [lo]. The electron is to some degree scattered backward and forward between the two reflecting regions corresponding, roughly, to the metal-vacuum interface at z = 0 and the top of the surface barrier at z, before leaving the metal. The corresponding electronic wave in the region between z = 0 and z, is therefore made up, to a small extent, by partially reflected waves which may be in phase (when z,, which is proportional to @, is approximately equal to an integral multiple of the average electronic wavelength) or out of phase, leading respectively to an increase or decrease in the amplitude of the wavefunction in the surface barrier region, and hence to a corresponding increase or decrease in the emitted current density. Obviously, a quantitative analysis of the deviations from the Schottky line must be based on eq. [ 121 which describes the current density as a function of applied field. It is clear that i(the average transmission coefficient) depends to some degree on the applied field. In evaluating i, according to eq. (14), the following simplifications are possible. The dependence of r(e, k,,), given by eq. (31), on the field comes exclusively from X, as defined by eq. (32a). From eqs. (61) and (59) we see that IX] = 1, and therefore r = 1, for

A. Modinos / Theory of thermionic emission

488

U-C 0, except in the immediate vicinity of the barrier maximum. On the other hand, for u > 0, 1h I= 0 except again in the immediate vicinity of the barrier maximum. The region around the barrier maximum where 1x1 is neither zero nor unity increases with the applied field, but even at the higher applied fields used in a typical experiment this region is only a small fraction of k,T (about one tenth or so) for TZ 1000 K. Hence only a small fraction of the emitted current comes from this energy region. It is for this reason that the deviations from the Schottky line are generally small and in fact negligible in the low field regions (see figs. 6 and 7). In this region (provided space charge limitation does not occur and that patchiness on the surface can be avoided; see, e.g., refs. [8] and [lo]) the experimental points will fall on a straight line whose slope equals m. By extrapolating this line to zero field we obtain and “extrapolated to zero field” current density given (see eq. (12)) by the well known Richardson equation J(0) =Ai(T,

0) exp(-$/k,T).

(76)

In the limit of zero field, eq. (61) gives IX, I= 1, therefore,

X, =O,

for u
6 4’) =

foru>O;

(77)

to eq. (31) we obtain

(p(c,k,,) I*, foru>O,

1, 1

for uC0.

(78)

In order to obtain Z(T, 0), we substitute eq. (78) into eq. (14), and noting that most of the contribution to the integral in eq. (14) comes from a very narrow region of energy (~5 k,T) and k,, space (k,, 5(2mkBT/h2)‘/*), we replace &t, k,,) over that region by p(O,O). Hence we obtain i(T,

0) =

1-

I/-@, O)]‘,

(79)

which shows (see eqs. (32b) and (69)) that the pre-exponential term in Richardson’s equation is determined entirely by the potential field inside the metal and the immediate vicinity of the metal-vacuum interface (I ~5 /?). It is worth noting that eqs. (77) have been obtained on the basis of the parabolic approximation to the potential in the region of the barrier maximum. This approximation is valid in the range of applied field used in a typical experiment and it is therefore legitimate to use the above formulae to obtain the “extrapolated to zero field” current density. This corresponds to the procedure by which this quantity is obtained in practice. However, we must emphasise that the parabolic approximation breaks down in the limit of zero field, and therefore eq. (77) does not give the reflection coefficient correctly just above the barrier (the vacuum level for zero field). Obviously the reflection coefficient equals unity at u = 0 + diminishing rapidly to zero as u increases. A measure of the deviation from the Schottky line is provided by the

A. Modinos / Theory of thermionic emission

following

quantity

489

[ 1,9,13]

AJ( F) = { ln[ J( F)/J(O)]

- mF’/‘}

/ln( lo),

(80)

where m is the theoretical slope of the Schottky line (eq. (75)) and J(0) is the value of the “extrapolated to zero field” emitted density. We expect that, in practice, the experimental points at relatively low applied fields will fall on a straight line - or closely about it - with a slope equal to m (within the experimental error). However, it is not certain that extrapolating to zero field the experimental Schottky line will give an experimental value of J(0) equal to the calculated one. The difference between the two values may be due to inaccuracy in the calculation of i(T, 0), but also in the assumed value of the workfunction $J at the given temperature (see eq. (76)). In plotting AJ( F) the experimentalist will naturally put J(O), in eq. (80) equal to its experimental value. (The corresponding curve, for the same set of data, but with J(0) in eq. (80) equal to its theoretical value will be displaced relative to the former but otherwise will be identical with it.) We emphasise that the deviation from the Schottky line as defined by eq. (80) contains a small monotonic component, besides a periodic one, due to the fact that as the field increases, some electrons with normal energy below the barrier maximum begin to contribute to the emitted current. The deviation from the Schottky line due to this tunneling contribution to the current is very small, for typical values of the applied field, but possibly of the same order of magnitude as the amplitude of. the periodic term. (see, e.g. figs. 6 and 7). Substituting eqs. (12) and (76) into eq. (80) we obtain AJ(F)=ln[i(T,

F)/i(T,O)]/ln(lO).

(81)

i(T, 0) is given by eq. (79). In order

l-r=l-

Vu)

+11(0,0>

1 +p(o,o)

to evaluate

i(T, F), we put

*

(82)

’

h*(u)

i.e. we replace ~(e, k,,) in eq. (31) by its value at z = 0, k,, = 0, as we have done in deriving eqs. (79) and for exactly the same reason. We obtain

A(u) +p(0,0) 1 +p(O

0) 9

j,‘(u)

1*] ev(-6)

du.

(83)

We have already noted that jhj = 1, in which case the integrand in eq. (83) vanishes, for u < 0 except in the immediate neighbourhood of u = 0, that X = 0 for u > 0 except again in the immediate neighbourhood of u = 0, and that the region where Ih 1is neither unity nor zero is very small (a fraction of k,T). The contribution to the integral from this region must in general be calculated numerically. The contribution to the integral from outside this region (where X = 0) is obtained analytically. We note that in earlier theories [9-121 of the periodic deviations from the Schottky line approximate analytic expressions

A. Modinos / Theory of thermionic

490

emission

for t‘
6. Numerical results for Cu(100) The phase shifts S, (1 = 0, 1,2) which describe scattering by an individual copper muffin-tin atom have been given in a parametrised form by Cooper et al. 1371. Using these phase shifts we caiculated the band structure of copper along the I’X symmetry direction (normal to the (100) plane) for three different values of the lattice constant a = 6.831 au, 6.996 au, and 7.054 au. The results are shown in fig. 3 by the solid, broken and dot-dash curves,

----_----

r

vacuum

level

k

=

12.05rV

(au)

X

Fig. 3. The band structure of copper along the TX direction for three different values of the lattice constant u. Solid curve: rr=6.83i au. Broken curve: tr=6.995 au. Dash-dot curve: u=7.054 au.

A. Modinos / Theon, of thermionic emission

491

respectively. We first note that the band-structure at zero temperature (a = 6.83 1 au) is essentially identical with that obtained self-consistently by Moruzzi, Janak and Williams [22]. Secondly, we observe that the occupied bands (below E, = 7.55 eV, the energy is measured from the constant potential between the muffin tin spheres) for the thermally expanded lattice, a = 6.995 au (T= 800 K) and a = 7.054 au (T= 1060 K), are practically identical with those at zero temperature (a = 6.831 au). We assume therefore that E, is practically independent of temperature. We further assume that the workfunction does not change significantly in the temperature region (800-l 100 K) of interest to thermionic emission and put it equal to $ = 4.5 eV (see, e.g., Haas and Thomas [20]). Accordingly the vacuum level lies 12.05 eV above the muffin-tin zero and in the middle of the energy gap, shown in fig. 3, which is approximately 4 eV wide. Obviously, in the absence of absorption (zero imaginary component of the potential) the reflection coefficient r(<, k,,) for electrons with c = 0 and k,, = 0 will be unity since the electron cannot penetrate into the metal, which in turn means a negligible thermionic current. This is not true in practice, for the imaginary part of the potential is not zero. It is obvious that careful measurements of the energy distribution of the thermionically emitted electrons and of the current itself (deviation from the Schottky line) when properly analysed can provide us with very useful information on the surface potential, both its real and its imaginary part. We performed calculations of the above mentioned quantities, assuming that the real and imaginary parts of the surface potential are given by eqs. (16) and (18) respectively, using different values of the parameters in these equations to test the dependence of the measurable quantities on these parameters. As far as the real part of the surface potential is concerned, we tried three different barriers A, B and C as follows (the meaning of D,,defined in section 3, and V(0 + ) is apparent from fig. 2): A:

D, =O;

Z ,,

B:

D,=O;

Zo =

c:

D, = 0.7 au;

z0 = 1.129 au.

=0.5645au 1.129 au

(V(O+)

=O),

(T/(0+)

=+(Er

(84) +$)),

(85) (86)

The parameters cr and y which determine the imaginary part of the potential inside the metal (eq. (19)) have been estimated by McRae [24] from optical data. His estimates are crEO.02 y=

1.7.

(v,,(E)

in eV),

(87a) (87b)

With the above choice of values Vi, at the vacuum level equals Q,,(e N 0) = 2.5 eV. McRae and Caldwell [21] obtained a value for j?, the parameter in eq. (18) which determines the range of the imaginary part on the vacuum side of the interface, by fitting very low energy (O-15 eV) data for the elastic reflection coefficient at a Cu(100) surface at room temperature to a theoretical curve for

A. Modinos ,I Themy of thermionic emission

492

the same quantity. The real part of the surface barrier, in their theory, is given by eq. (16) with F= 0 and with values of the parameters given by eq. (84). With the imaginary part of the potential given by eqs. (18) and (87) they found that in order to fit the data to their theory, p must be equal to /3-2au.

(88)

We note that in their calculation the diffraction by the metal was treated approximately (two-beam approximation) and that may have introduced some error in their calculation. Also the fact that in their calculation the field is zero, whereas in our case it is not, may make some difference in the value of the calculated elastic reflection coefficient (the applied field removes the coulombic tail from the surface potential). In any event, using their choice of parameter values (eqs. (84) and (87)), we found that the elastic reflection coefficient at E N 0 equals approximately 0.69 which is larger than their value (0.57) by approximately 18%. Since the experimental evidence [20,21] suggests that the reflection coefficient at E = 0 is about 0.6 and is to a first approximation independent of temperature, we decided to treat in our calculations p as an adjustable parameter. In order to examine the dependence of the measurable quantities on the temperature we performed calculations for two different temperatures T= 800 K and T = 1060 K. The temperature dependent phase shifts at the two temperatures were calculated on the basis of eq. (56) using the parametrised zero temperature phase shifts of Cooper et al. [37] and an effective Debye temperature

O=245K,

(89)

which is roughly equal to 2/3 of the Debye crystal (see comments in section 4.3). Table I Dependence temperature

IMOY0)l ar&(O, 0))

i(T.0)

of ~(0.0)

and

of r( T, 0) on the parameters

temperature

in the bulk

of the optical

potential

and

on the

I

II

III

IV

V

VI

VII

0.832 2.033 0.308

0.781 2.033 0.390

0.8165 2.017 0.333

0.762 2.200 0.419

0.758 3.055 0.425

0.750 2.198 0.4376

0.762 2.104 0.419

T=8OOK,r,=0.5645 au, ~,=O.O. V,,,,(r=O)=-0.253 eV,p=2.0au. T=8OOK, +,=0.5645 au, O,=O.O, V,,(c=O)= -0.253 eV. p=3.5 au. T=800 K, z0 =0.5645 au, D,=O.O, V,,,,(c=O)= -0.253 eV. p=3.5 au; the thermal of the atoms has not been taken into account. IV: T=8OOK,z,=1.129 au, D,=O.O, V,,(r=O)=-0.253 eV, p~3.5 au. T=8OOK,r,=1.129 au, 0,=0.7 au, V,,(r=O)=-0.253 eV, p=3.5 au. V: VI: T=8OOK,z,=l.129 au, O,=O.O, V&c=O)=-0.4eV,/?=2.Oau. VII: T= 1060 K, z,, =0.5645 au, 0, =O.O, Vi,,,(c=O): = -0.253 eV, p~3.5 au.

I: II: III:

of the

vibration

A. Modinos /

Theoty of thermionic emission

0.3

493

; Y E .d 7

0.2

0.3

I 0.044

I

I

I

I

I

0.844

0.441 E

(CV)

Fig. 4. The enhancement factor R(c) as defined by eq. (90) for two temperatures: T= 800 K (curve II); T= 1060 K (curve VII). Curve III was obtained with the thermal vibration of the atoms neglected (other parameters as for II). The dash-dot curve gives the variation of the imaginary component of the potential with the energy.

The results of our calculations are summarised in table 1 and in figs. 4 to 8. In table 1 we show how the transmission coefficient i( T, 0), defined by eq. (79) depends on the parameters of the calculation. We see from this table that the magnitude of i( T,0)is basically determined by the magnitude of the imaginary part of the potential at the vacuum level &,(E = 0) (different values of this quantity correspond to different values of (Y in eq. (19); we kept the other parameter in this equation constant y = 1.7), and by the extension of this potential (parameter /I) into the vacuum region. The set of parameter values denoted by I are those of McRae and Caldwell [21]. The corresponding transmission coefficient (0.308) is smaller than the experimental value (0.4-0.45) [20,21]. Choosing a larger value, j3 = 3.5 (sets III, IV and V), leads to values of

A. Modnos

494

/

Theo9

of thermionic

emission

w

a 0.4 -

Fig. 5. Dependence of the enhancement R(t) on the parameters of the optical potential. various curves correspond to different sets of parameters values as described in table I.

The

i(T, 0) in the region 0.39-0.42 depending on the values of z0 and-D,. It is clear that the latter two parameters do not affect the magnitude of t( r, 0) in any significant way. We note (see column VI) that the larger transmission coefficient can also be obtained by keeping p = 2 au increasing instead the value of V,,( E = 0), i.e. by increasing the value of LYin eq. (19). (We note however that ideally cx being a bulk parameter must be determined by independent experiments.) The result under III was obtained with the same parameter as in II, but with the atoms fixed at their mean positions (thermal vibrations neglected). We see that the thermal vibration of the atoms reduces the reflection coefficient (1 - i) by less than 10%. We remember that our calculation (see section 4.3) overestimates this reduction, hence we may conclude that the effect of thermal vibration on electron reflection at the surface in the limit of zero energy (C = 0) is relatively small compared to that of electron-electron collisions (represented by the imaginary part of the potential) which is independent of the temperature. For this reason, i(T, 0) at T = 1060 K (result VII) is only 7% larger than the corresponding quantity at T = 800K. Figs. 4 and 5 show the results of our calculation for the total energy

A. Modinos / Theory of thermionic emission

495

Fig. 6. The deviation from the Schottky line. Its dependence on temperature and on the range of the imaginary component of the potential outside the metal. The three curves correspond to different sets of parameter values as described in table 1.

distribution of the emitted electrons for the different sets of parameters (I, II, etc.) as defined in table 1. For obvious reasons we have chosen to plot the enhancement factor R(E) defined by NC)

=jM/j&),

(90)

where j,,( 2) is the .“free-electron” distribution defined by eq. (15) rather than j(e). We note that in calculating j(c), according to eqs. (9) and (31), we have taken into account the variation of p with E and k,,. (Although most of the current comes from the region e < k,T, k,, = 0, the energy distribution can be measured over a much wider energy region, 1 eV and perhaps more, and over that region ~1 may change significantly.) In fig. 4 we show R(c) for two different temperatures: T= 800 K (II) and T = 1060 K (VII). Curve III, obtained with the same parameter values as curve II except that the thermal vibration has been disregarded, shows that the latter does not affect the shape

496

0.00

c,

A.

Modinos

/

Theon,

of thermionic

enrissiorl

-

-

a 0.04 -

Fig. 7. The deviation from the Schottky line. Its dependence on the surface curves correspond to different set of parameter values as described in table I.

barrier.

The three

of R(E) in any significant way. The dash-dot curve on the same figure shows the variation of the imaginary part of the potential with energy when y = 1.7 (see eq. (19)). It is clear that in this instance the variation of R(E) (curve II) derives to a large extent from the corresponding variation of F,(E) with the energy. Note, however that R(E) increases faster than Vim(e), presumably because as the energy moves away from the centre of the gap (see fig. 3.) the “incident” electron penetrates further into the metal and therefore the probability of it being scattered inelastically (absorbed) increases. At energies approaching the edges of the gap the penetration of the electron into the metal, and therefore absorption, may increase considerably. This is demonstrated quite clearly by curve VII obtained at the higher temperature. The (assumed) lattice constant for this temperature leads to the band structure shown by the dotted curve in fig. 3. It is seen that the upper edge of the gap has been lowered so that electrons with c = 1 eV (E = 13.0 eV) are now nearer to the edge of the gap, they penetrate more into the metal, absorption increases considerably with

Fig. 8. barrier

A. Modinos / Theory of thermionic emission

497

The deviation from the Schottky ‘line for an assumed value of p(O,O)=O.218+0.2202, B (eq. (85)); (T= 800 K).

for

a consequent decrease in the reflection coefficient which leads to a correspondingly large increase in R(c). Fig. 5 shows how R(E) depends on the various parameters of the optical potential. The remarks we have made in relation to table 1 apply equally well here. The numerical results on the deviation from the Schottky line are summarised in figs. 6 and 7. The three curves in fig. 6 correspond to the set of parameters values denoted by I, II and VII in table 1. Comparing curves I and II we see that increasing the range of the imaginary component of the potential outside the crystal leads to a decrease in the amplitude but has no effect on the “period” or the phase of the deviations from the Schottky line. The same holds true as to the dependence of these deviations on the magnitude of the imaginary component. An increase in the latter leads to a decrease in the amplitude of the deviations but has no effect on the period or the phase of these deviations. Comparing curve II and VII we see that an increase in the temperature has no effect on the period and phase of these deviations but leads again to a decrease in the amplitude of these deviations. This result is in

498

A. Modinos / Theory of thermionic emrssion

agreement with observations on polycrystalline emitters [ 1,8]. Fig. 7 demonstrates the dependence of the deviation from the Schottky line on the parameters (zc and 0,) which determine the real part of the potential. The three curves in this figure correspond to the set of parameter values denoted by II, IV and V in table 1. It is seen that a change in za, with D, kept constant, leads to a noticeable change in both the phase and the amplitude of the deviation especially at the higher applied fields (compare curves II and IV) but when both D, and z0 are changed the corresponding changes in the deviation are much less (compare curves II and V). It is worth noting at this point that the available experimental data, those on polycrystalline emitters of molybdenum, tungstent and tantalum (see, e.g., fig. 3 in ref. [12]) and the data for tungstent (111) [ 161, all show a maximum at about @ = 0.035 (V/A)‘/*. On the other hand, most of the earlier calculations (in these calculations energy-bandstructure effects and inelastic collisions are entirely neglected, and the surface potential barrier. is represented by a simple image potential barrier) give maxima of about 0.03 (V/A)‘/* (see, e.g., ref. [13]). It would seem to us that there is no obvious reason for the phase of the deviation to be the same for different emitting planes and, in this respect, more experimental data from single crystal planes are required to establish whether or not this phase depends, and to what degree, on the specific properties of the emitting plane. At this stage we can only say that if the bulk parameters are known from independent experiments the range of the imaginary component of the potential can be determined from measurements of the energy distribution and of the current density (the magnitude of i(T, 0) and the amplitude of the deviations from the Schottky line suffice to determine /3) and that, whereas we expect to be able to reproduce the experimentally observed phase and “period” of the deviations by an appropriate choice of z0 and D,, these parameters cannot be determined uniquely (as is clear from fig. 7) from such data. Only if additional information on these parameters is available from independent experiments, e.g. field emission experiments [38,39], can z0 and D, be uniquely determined. We must also emphasise that representing the potential in the region of the metal-vacuum interface by eqs. (16) and (18) is in itself an approximation. Obviously, if we choose D, = 4 au, there is no doubt that the potential for z > 0 (see fig. 2) will be adequately described by eq. (16) but we cannot, of course, assume that the actual potential between the muffin-tin spheres of the top layer and the plane z = 0 is constant or, for that matter, that it is a function of z only. In principle the potential in this region, at least its real part, must be calculated selfconsistently. The same applies to the muffin-tin potential of the top layer which may be different from that in the bulk of the crystal. Were the potential in the interface region (between the muffin-tin spheres of the top layer and the plane z = 0) known from such calculations, we could, at least in principle, take its effect into account in our evaluation of p,,, by treating this region as an additional “scattering layer” in a manner analogous to that described in section 4.3. In the meantime the best one can

A. Modinos / Theo?

of thermionic emission

499

hope for, from an effective potential, such as the one described by eqs. (16)-( 19), is that for particular values of the parameters this potential scatters the electron in the interface region in the same manner as the more realistic self-consistent potential does. We have already mentioned that in the earlier theories of the “periodic” deviation from the Schottky line [9- 121 the assumption is made that ]p(O, 0) 1in eq. (82) is small, i.e. (~(0, O)] K 1, and that under this condition it is possible to obtain an approximate analytic formulae for i(T, F) and AJ( F) in which p(O,O) appears as a parameter. The reader will find the relevant formulae in the cited papers. Here we would like to point out that when jp(O, O)] a 1 the deviations from the Schottky line are to a good approximation symmetric about the Schottky line, which is not true for the larger values of ]p(O, O)] appropriate to Cu(100) (given in table I), as is obvious from figs. 6 and 7. In order to demonstrate the above, we calculated i(T, F) on the basis of eq. (83) for an assumed value of ~(0, 0) = 0,218 + 0.22Oi, for barrier B (eq. (85)). The results for the corresponding AJ(F) are shown in fig. 8. In this case the deviation is practically symmetric about the Schottky line but the amplitude is much smaller in agreement with the earlier theories.

References [I] G.A. Haas and R.E. Thomas, in: Techniques in Metals Research (Wiley, New York, 1972) Vol. VI. p. 91. [2] O.W. Richardson, Proc. Cambridge Phil. Sot. I I (1902) 286; Phil. Mag. 12 (1912) 263; W. Schottky, Physik 2. IS (i914) 872: 14 (1923) 63; M. von Laue, Jb. Radakt. Efektr. 15 (1918) 205: 301. [3] S. D&man, Phys. Rev. 21 (1923) 623. [4] R.H. Fowler and L.W. Nordheim, Proc. Roy. Sot. (London) Al I9 (1928) 173; L.W. Nordheim, Proc. Roy. Sot. (London) Al21 (1928) 626. [5] A. Sommerfeld, Physik Z. 47 (I 928) I. [6] R.L.E. Seifer! and T.E. Phipps, Phys. Rev. 56 (1939) 652. [7] E. Guth and C.J. Mullin. Phys. Rev. 61 (1942) 337. 181 W.B. Nottingh~, in: Handbuch der Physik, Vol. XXI (Springer, Berlin, 19.56) P. 1. (91 D.W. Juenker, G.S. Colladay, Jr. and E.A. Coomes, Phys. Rev. 90 (1953) 772; D.W. Juenker, Phys. Rev. 99 (1955) 1155. [IO] C. Herring and M.H. Nichols, Rev. Mod. Phys. 21 (I 949) 185. [I I] SC. Miller and R.H. Good, Phys. Rev. 92 (1953) 1367. [ 121 P.H. Cutler and J.J. Gibbons, Phys. Rev. I I I (1958) 394. [ 131 G.G. Belford, A. Kupperman and T.E. Phipps, Phys. Rev. 128 (1962) 524. (141 M.H. Nichols, Phys. Rev. 57 (1940) 297. [ 151J.C. Riviere, Solid State Surface Science (Dekker, New York, 1969) Vol. 1, p. 179. [16] D.F. Stafford and A.H. Weber, J. Appl. Phys. 34 (1963) 2667. [I71 A.R. Hutson, Phys. Rev. 98 (1955) 889; H. Shelton, Phys. Rev. 107 (1957) 1553. [ 181 J.W. Gadzuk and E.W. Plummer, Rev. Mod. Phys. 45 (1973) 487. [ 191 R.G. Wilson, J. Appl. Phys. 37 (1966) 37. 1201 G.A. Haas and R.E. Thomas, J. Appl. Phys. 48 (1977) 86.

500

A. Modinos / Theon, of thermionic emission

[21] E.G. McRae and C.W. Caldwell, Surface Sci. 57 (1976) 766. [22] V.L. Moruzzi, J.F. Janak and A.R. Williams, Calculated Electronic Properties of Metals (Pergamon, New York, 1978). [23] J.B. Pendry, Low Energy Electron Diffraction (Academic Press, London, 1974). [24] E.G. McRae, Surface Sci. 57 (1976) 761. [25] L.D. Landau and E.M. Lifshitz, Quantum Mechanics (Pergamon, London, 1958). [26] J.L. Beeby, J. Phys. Cl (1968) 82. [27] J.B. Pendry, J. Phys. C4 (1971) 2501. [28] S.Y. Tong, T.N. Rhodin and R.H. Tait, Phys. Rev. 8 (1978) 421. [29] P.P. Ewald, Ann. Physik (4) 64 (1921) 253. [30] K. Kambe, 2. Naturforsch. 22a (1967) 322, 422. [3 1) D. Tabor and J.M. Wilson, Surface Sci. 20 (I 970) 203. [32] C.B. Duke and G.E. Laramore, Phys. Rev. B2 (1970) 4783. [33] B.W. Holland. Surface Sci. 28 (1971) 258. [34] R.J. Glauber, Phys. Rev. 98 (1955) 1692. [35] E.T. Whittaker and G.N. Watson. A Course of Modem Analysis (Cambridge University Press, 1952). [36] E. Jahnke and F. Emde, Tables of Functions (Dover, New York, 1945). [37] B.R. Cooper, E.L. Kreiger and B. Segall, Phys. Rev. B6 (1971) 1734. [38] A. Modinos, Surface Sci. 70 (1978).52. [39] A. Modinos and G. Oxinos, J. Phys. C9 (1981) 1373.